Hmm… I’d like to have both – the overlay button AND the clickable image. That way if people are on a mobile or just want to click the image before the buttons pop up (or if they want to click outside the button area) – it still takes them to the product.

You should be able to make the change to the up woocomerce-config>config.php I posted above and then it will work as you describe. The css is what hides the buttons and the php change is what adds the link to the whole image.